## Arithmetic moduli of elliptic curves.(English)Zbl 0576.14026

Annals of Mathematics Studies, 108. Princeton, New Jersey: Princeton University Press. XIV, 514 p. hbk: $78.00; pbk:$ 22.50 (1985).
This book gives a full account of the theory of moduli for elliptic curves order $${\mathbb{Z}}$$ from a scheme-theoretical point of view. In order to give representable moduli problems elliptic curves need to be rigidified by introducing supplementary structures not preserved by the automorphism group. The $$\Gamma$$ (N)-problem, or full level-N problem, is determined by a choice of generators of the group of division points with order dividing N. For elliptic curves over an arbitrary base ring this has to be stated in terms of subgroup schemes exactly generated by two divisors. There are similar statements for the $$\Gamma_ 0(N)-$$, $$\Gamma_ 1(N)$$-problems and their variants. The authors introduce functorial machinery to treat these problems in a uniform way and avoid algebraic stacks by the device of relative representability.
The first four chapters give a self-contained account of the elementary theory of elliptic curves over a scheme, their finite subgroup and moduli problems. The reader is referred to other sources for deeper results such as the Serre-Tate theorem on deformations and the construction of Tate curves.
The subsequent chapters deal with the intricate details involved in completing the moduli scheme at infinity and describing its structure at points corresponding to super-singular curves. In particular two chapters cover problems specific to characteristic p and Igusa curves. The culminating result is a general good-reduction theorem for Jacobians.
Altogether this book is an essential complement to the many works on the transcendental theory of elliptic curves.
Reviewer: G.Horrocks

### MSC:

 14H10 Families, moduli of curves (algebraic) 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 14D22 Fine and coarse moduli spaces 14H52 Elliptic curves 14D20 Algebraic moduli problems, moduli of vector bundles 11F03 Modular and automorphic functions 14H45 Special algebraic curves and curves of low genus
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